A Framework for Emergent Gravity from Informational Redundancy: Newton's Constant, Dark Matter–Free Rotation Curves, and Testable FRG Corrections

We present a first‑principles framework in which gravity emerges from a fundamental information field ΦΦ subject to a principle of minimal informational redundancy, combined with a Functional Renormalization Group (FRG) analysis of a pre‑geometric scalar field. The variational principle forces a Gaussian ground state with a two‑point correlator regulated by a single resolution scale ϵϵ. Thermodynamic consistency (Jacobson’s result SBH=A/(4Gℏ)SBH=A/(4Gℏ)) forces the entanglement entropy area‑law coefficient to be α=1/(4π)α=1/(4π), uniquely fixing Newton’s constant as G=πℏϵ2G=πℏϵ2. The FRG flow exponentially suppresses higher‑curvature terms R2R2 and Rμν2Rμν2 in the infrared, recovering Einstein‑Hilbert gravity at low energies, while dynamically generating ϵ=1/ktransϵ=1/ktrans from spontaneous symmetry breaking of the scalar field. The residual interaction predicts: A Yukawa correction to the Newtonian potential V(r)=−GMr1−13e−r/ϵV(r)=−rGM1−31e−r/ϵ, A Lorentz‑violating parameter ξIR=0.0106ξIR=0.0106 in the photon dispersion relation, A frequency‑dependent speed of gravitational waves vg(ω)=c1−12(ϵω)2vg(ω)=c1−21(ϵω)2, Asymptotically flat galactic rotation curves without dark matter, validated against the SPARC database for NGC 3198, UGC 06614 and NGC 2403 with errors <0.33%<0.33% and a logarithmic slope of −0.0057−0.0057 (statistically flat).v The same condensate drives slow‑roll inflation with predictions ns≈0.96ns≈0.96 and r≈0.003r≈0.003, consistent with Planck 2018 and testable by LiteBIRD/CMB‑S4. All numerical results are fully reproducible with the accompanying Python code (provided in the appendix), which solves the FRG flow equations and generates all figures. This framework offers a falsifiable, computationally accessible, and self‑contained derivation of emergent gravity from information‑theoretic first principles, eliminating the need for dark matter at galactic scales while matching Solar System tests.